The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

I was especially curious how well the Common Core State Standards for Mathematics (hereafter, CM) kept these rather sweeping promises, a document of some 90 pages that covered mathematics concepts expected to be taught in K-12. However, I focused only on concepts related specifically to logic, a subject I taught at the college level for several years, and to which I made original contributions.

Does the CM document provide mathematics students “a consistent, clear understanding” of basic concepts of logic? Does the document provide logic-related “knowledge and skills” mathematics students will need “for success in college and careers”? I explain why the answer to both questions is NO and present an alternative. American students will not be “fully prepared for the future” without understanding basic logic.

Background

It may be objected that I am being unfair (illogical?) in criticizing mathematics standards for failing to do a good job of explaining logic. After all, what does logic have to do with mathematics? Even if logic is relevant to mathematics, how exactly is it relevant? And why is it important that our schools make sure K-12 mathematics students have a firm grasp of the concepts of logic? Readers who already know the answers can skip ahead.

It has been understood at least since the time of Euclid (around 300 BC) that mathematics is special in content as well as method. Mathematical results have seemed to possess the highest degree of certainty possible, likewise the methods by which they are achieved. Thus, one of the most famous books in the history of mathematics, Euclid’s Elements, starts out with a small set of assumptions (axioms) along with defined and undefined terms, and then derives a wealth of interesting facts (theorems) about plane geometry.

Logic comes into play because it studies what “derives” means. Here are three questions logic seeks to answer that explain its relevance to mathematics. These questions are of general interest as well:

(1) What does it mean to derive one mathematical statement from another?

(2) What is the difference between correct and incorrect derivations?

(3) Are there methods for testing the correctness of derivations?

Readers will not find Euclid’s Elements asking these questions, let alone providing answers. Countless other mathematics texts published over the centuries have not done any better. I recall a conversation years ago with my mathematics adviser at the City College of New York, Fritz Steinhardt, who told me the questions were “philosophical.” I changed majors to philosophy.

Why is logic taught in philosophy departments? The reason is Aristotle (384-322 BC), who is credited with getting logic started. Unfortunately, Aristotle’s syllogistic analysis, though a good first approximation, turned out to be of limited value in mathematics. For example, Euclid’s famous reductio ad absurdum proof that the square root of 2 is not a rational number requires far more powerful methods to analyze than the syllogism, as do many other results in mathematics, a fact that became even more evident as the subject grew far beyond Euclid. Powerful analytical methods became available only late in the 19th century thanks to the work of the German mathematician Gottlob Frege (1848-1925) — a name obscure to most people.[ ] Modern logic was finally able to answer (1)-(3) and mathematics has benefited significantly, even if a bit late in the day. Logic also proved critical in the development of computers, their hardware as well as software.

Logic in the CM Document

Evidence occurs as early as p. 7 that CM document authors — who are not named, though there is a “Sample of Works Consulted” at the end — seemed to agree on the relevance of logic to mathematics and that mathematics standards should cover logic concepts:

Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and — if there is a flaw in an argument — explain what it is.

Moreover, three of the eight Mathematical Practices (MPs) for all levels — including kindergarten — mention what appear to be logic-related skills:

Logic-related skills are also implicit in such student taskings as (and I quote):

Make plausible arguments; decide whether arguments made sense; explain flaws in an argument; use informal arguments; clarify or improve arguments; determine domains to which an argument applies; respond to the arguments of others; listen [to] or read the arguments of others; justify conclusions; build a logical progression of statements to explore the truth of conjectures; explain the reasoning used; draw conclusions; draw inferences about a population; draw valid conclusions about the whole population; draw informal comparative inferences; distinguish correct reasoning from that which is flawed.

What’s the Problem?

The high school algebra section states (p. 62, my italics):

An equation can often be solved by successively deducing from it one or more simpler equations. [T]he formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using [a] deductive process. Inequalities can be solved by reasoning about the properties of inequality.

Aside from the fact that students should not have to wait until high school to learn that reasoning in mathematics is deductive, on p. 7 of the CM document we find the phrase “reason inductively about data” but, oddly, nowhere in the document do we find the term “reason deductively” or its equivalents. In fact, mathematical reasoning is deductive whether it is abstract or quantitative (MP2). It is the deductive reasoning of other students that is to be critiqued (MP3). It is regularity in repeated deductive reasoning that mathematics students should look for (MP8). This should have been made clear right away, followed by an explanation of “deductive”; “inductive” is also left unexplained.

What bothered me a lot more, however, was the repeated use of non-standard, fuzzy terminology to describe a key attribute of deductive arguments already understood in logic. Instead of characterizing such arguments using the standard terms “valid” and “invalid” and then explaining what they mean, the CM document has students consider whether arguments are “plausible,” “effective,” “viable,” or “make sense.” The “viable” characterization occurs in the third Mathematical Practice (MP3), which the document requires teachers to implement in all of K-12 but never explains!

Logic does not study whether arguments are plausible, effective, viable, or make sense, for reasons Frege noted long ago. These terms suggest that the logical correctness of an argument is a matter of personal opinion — or, to use the current favorite, “choice.” Thus, the CM document creates the impression that students could submit — and teachers accept — test answers based on arguments that are plausible, effective, viable, or makes sense “to them” — even though the arguments are in fact invalid. Subjectivism and fuzzy thinking have no place in mathematics or anywhere else in school for that matter.

The CM document appropriately notes that reasoning can be “correct” as well as “flawed” (p. 7). An explanation of the difference is not provided, however, leaving the matter open to interpretation; nor do we find insistence that only correct reasoning is acceptable in mathematics. Using fuzzy notions of arguments (and reasoning) being “plausible,” “effective,” “viable,” or “making sense,” might allow teachers to pass mathematics students who did not reason correctly. The absence of logic standards (!) means teachers could “choose” to assess student reasoning in California one way and another way in Florida, both wrong. This is not at all the path to “a consistent, clear understanding” of a concept at the very heart of mathematics. It is the path to chaos.

A Solution Outlined

I can indicate only briefly and informally how validity in arguments might be usefully taught to mathematics students. Details are in logic books, including mine.

Students should be made aware that validity has two forms. For ease of reference, let us call them A-validity and B-validity. (How they are related is an interesting question.)

An argument is A-valid just in case under no interpretation are the premises true and the conclusion false; otherwise the argument is A-invalid.

An argument is B-valid just in case there is a derivation of its conclusion from the set of its premises. (B-invalidity is more complicated.)

A-validity should be taught first. Students can reflect on the meaning of premises and conclusion in an argument and try to find ways that the premises could be true and the conclusion false. This will give them a “feel” for the strong bond between the elements of an A-valid argument. Formal methods such as truth tables can be taught later.

B-validity is useful in teaching mathematical proofs, where premises are all theorems. Students must learn how to find the relevant theorems, which assumes mathematical knowledge. Then they must find the rules of logic that will lead from theorems to conclusion, which assumes knowledge of rules of logic and how to apply them.

In a mathematical proof, it is rules of logic that justify each step of the reasoning process. Correctly applied, they move established mathematical knowledge slowly but surely until the end is reached and new knowledge is created. Rules of logic can be programmed into computers to carry out or check mathematical proofs.

In the early stages of teaching proofs steps should not be skipped — even if “obvious” — and rules of logic justifying each step should be stated. Students can get lost if steps are skipped and may become discouraged from learning a critical mathematical skill. They cannot be expected to figure out rules of logic on their own.

The Bottom Line

States that have already adopted Common Core Standards for Mathematics should urge teachers to introduce students to logic concepts as early as possible — not the vague and arbitrary ones in CM but the real thing that will enable students’ “inference engines” to work properly. CM “fuel” will cause those engines to sputter and stall.

Arnold Cusmariu received his Ph.D. in philosophy from Brown University and has published on a variety of technical subjects. He recently completed a book titled Logic for Kids that introduces children to the basics of deductive reasoning. He can be reached at aclogic1@yahoo.com.